= Solution
This is the <dephased Bell-state mixture> with coherence parameter $2x$. The <Pauli correlation tensor> is
$$
T_{ij}=\operatorname{Tr}[\rho(x)\,\sigma_i\otimes\sigma_j],
\qquad T=\operatorname{diag}(2x,-2x,1),
$$
so spin measurements along unit vectors have $E(\mathbf a,\mathbf b)=\mathbf a^TT\mathbf b$. Choose
$$
\mathbf a=\mathbf e_z,\qquad \mathbf a'=\mathbf e_x,\qquad
\mathbf b=\frac{\mathbf e_z+2x\mathbf e_x}{\sqrt{1+4x^2}},\qquad
\mathbf b'=\frac{-\mathbf e_z+2x\mathbf e_x}{\sqrt{1+4x^2}}.
$$
Then $E(a,b)-E(a,b')=2/\sqrt{1+4x^2}$ and $E(a',b)+E(a',b')=8x^2/\sqrt{1+4x^2}$. The <CHSH inequality> expression is therefore
$$
\boxed{S(x)=2\sqrt{1+4x^2}>2\quad\text{for every }0<x\leq\tfrac12.}
$$
These axes also attain the <CHSH optimum from the correlation tensor>, since the two largest <eigenvalues> of $T^TT$ are $1$ and $4x^2$. At $x=1/2$ the value is $2\sqrt2$; at $x=0$ the explicit <LHV model> in part (c) applies. Thus \b[exactly $0<x\leq1/2$ has correlations incompatible with a local hidden-variable description].
Arbitrarily small positive coherence suffices in this particular one-parameter family, with optimally chosen axes; the violation above two tends to zero quadratically as $x\to0$. This does not imply that every entangled mixed state violates a <CHSH inequality>.
Back to article page
